Persiflage

(Now with updates!)

Let’s recap from part I. We have a Shimura variety \(Y\), a minimal projective compactification \(X\), and a (family of) smooth toroidal compactifications \(W\). We also have Galois representations of the correct shape associated to eigenclasses in

\(H^0(X^{\mathrm{ord}},\xi_{\rho}).\)

So at this point (well, not only at this point) there is some confusion. In the construction above, I am imagining that we are working with the rigid analytic space corresponding to the ordinary locus. But now there are some remarks in my notes about dagger spaces. Here is what I am imagining is going on. For any sufficiently small radius, we may consider the rigid analytic space \(Y[\nu]\) which corresponds (on the moduli level) to the appropriate abelian varieties \(A\) (with polarization and level structure and enomorphisms, blah blah) together with a canonical subgroup which (under some measure) is close to being ordinary. Then there is a “dagger space” \(Y^{\dagger}\) which is the limit of all such spaces. The issue (for me) is that I don’t really know anything about dagger spaces, but since this is probably not the main point, I will (again) elide the issue here. Of course, the goal is to realize the eigenvalues of the Eisenstein series \(\Pi\) inside this cohomology. Let’s assume that \(\Pi\) actually has good reduction at \(p\). Then it is probably going to be true that \(\Pi\) actually has finite slope, and so it lives inside the cohomology of some overconvergent neighbourhood of \(X^{\mathrm{ord}}\). So there’s some flexibility with exactly what spaces one is working with. Perhaps working with finite slope eigenforms might help to get local-global compatibility at \(p\).

(update: it’s most natural to work with the dagger spaces (whose cohomology is as described above) since that most naturally corresponds to the rigid cohomology groups occurring below.)

OK, so, we may take the direct limit over all compact subgroups \(U\) of the cohomology above, and we want to realize the Eisenstein series \(\Pi\) as a p-adic cusp form inside this space.

To this end, one introduces the following cohomology groups:

\( H^*_{c,\partial}(\overline{X}^{\mathrm{ord}}) := \mathbb{H}^{*}(W^{\mathrm{ord}}, \Omega^{\bullet}_{W^{\mathrm{ord}}}(\log \infty) \otimes
\mathcal{L})\)

OK. So this is just a definition, it isn’t supposed to obviously be functorial: we are taking the special fibre, lifting to characteristic zero, taking a toroidal compactification, then looking at the hypercohomology of the de Rham complex with log poles at the boundary. Well I guess one can do whatever one wants, I suppose.

So what is this? The hycohomology of the de Rham complex of a smooth variety \(M\) with log poles along some divisor \(D\) with normal crossings should just be the Betti cohomology of the complement of \(D\) in \(M\). The factor \(\mathcal{L}\) is the difference between the sub-canonical and canonical extensions, not entirely sure why it is there, presumably for some fundamentally important reason. So morally, I think the RHS should be computing something like the Betti cohomology of \(Y^{\mathrm{ord}}\), with the proviso that these are dagger spaces, not smooth complex varieties. So one should think of the LHS is some type of algebraic Betti cohomology of the ordinary locus.

(update: the remark about the Betti cohomology of the complement of \(D\) is correct, but the presence of the boundary divisor \(\mathcal{L}\) is exactly what, in the classical sense, changes the answer from the cohomology of the open variety to the interior cohomology. So the cohomology is somehow compactly supported towards the boundary of \(W\), but not the “other” part of the boundary (that is, the difference between \(W\) and \(W^{\mathrm{ord}}\).)

Let’s write down a spectral sequence:

\( H^i(W^{\mathrm{ord}}, \Omega^{\bullet}_{W^{\mathrm{ord}}}(\log \infty) \otimes
\mathcal{L}) \Rightarrow H^{i+j}_{c,\partial}(\overline{X}^{\mathrm{ord}}),\)

The existence of this spectral sequence must be a formal consequence of the definition and properties of hypercohomology. Note that the \(\Omega^j_{W^{\mathrm{ord}}}(\log \infty)\) are canonical automorphic sheaves of the standard type, so with the boundary piece \(\mathcal{L}\) the LHS consists of terms of the form \(H^i(W^{\mathrm{ord}},\xi^{\mathrm{sub}})\). To compute these terms, one can push foward via the map \(\pi: W \rightarrow X\) from the toroidal compactification to the minimal one. Then one notes that:

1. The higher direct images \(R^i \pi_* \xi^{\mathrm{sub}}\) vanish.
2. Since \(X^{\mathrm{ord}}\) is affinoid, its higher cohomology also vanishes.

The second point seems reasonable, I have no idea why the first is true. It is probably a really key point, which I might talk about in part III (note: RLT said nothing about this and there is no pre-print, so I have no idea how to prove this at the moment). Apparently it is important that one uses the subcanonical extension here. This implies that every class which occurs in the RHS in this new cohomology actually occurs in an \(H^0\) term on the LHS. Now one has Galois representations of terms of the form \(H^0(W^{\mathrm{ord}},\xi^{\mathrm{sub}})\), by the first construction – here it must be OK to pass between \(W\) and \(X\) using the Kocher principle. So we are reduced to showing that \(\Pi\) contributes to this new cohomology \(H^{\bullet}_{c,\partial}(X^{\mathrm{ord}})\).

(update: here is some more about higher direct images. Let’s say a little bit about what the toroidal compactifications look like. Let’s even imagine we are working with \(\mathcal{A}_2\) and are looking at a cusp where one has purely toric reduction. For the purposes of computing the higher direct images all that matters is the formal completion of \(W\), which at the boundary looks something like \(Z/\Gamma\) for some toric variety \(Z\) which is not of finite type. One shows that \(H^i(Z,\mathcal{O}_Z) = 0\) using Cech cohomology for \(i > 0\), which allows one to think of \(Z\) as contractible. Then one would like to say that \(H^i(Z/\Gamma,\mathcal{O}_Z)\) is also zero, which comes down to understanding the action of \(\Gamma\) on \(H^0(Z,\mathcal{O}_Z)\). Roughly one would like to say that \(\Gamma\) acts with no fixed points and and use Shapiro’s Lemma. Back to the specific example, one finds that \(H^0(Z,\mathcal{O}_Z)\) corresponds to positive semi-definite \(2 \times 2\) matrices, and \(\Gamma\) a finite index subgroup of \(\mathrm{GL}_2(\mathbf{Z})\). Here one should be reminded of the \(q\)-expansions of Siegel modular forms at the cusp — recall that \(q\)-expansions are given in terms of such matrices whose coefficients are invariant under \(M \mapsto X M X^{T}\). This action is free as long as \(\det(X) \ne 0\); at the level of \(q\)-expansions this corresponds exactly to working with cusp forms; this is why working with the sub-canonical extension allows one to restrict the positive definite forms on which the action is indeed free. In the degenerate case when \(n = 1\), then \(\Gamma\) is trivial, and so it even acts freely on the non-cusp form \(1\), which is why it doesn’t matter in that case.)

Note: the spectral sequences above is, like the Hodge-de Rham spectral sequence, a 1-st page spectral sequence. Thus the vanishing above does *not* imply that it degenerates. Moreover, it certainly won’t degenerate, since the RHS will turn out to consist of finite dimensional vector spaces, whereas the terms on the LHS are certainly not (as they are spaces of p-adic or overconvergent forms). (Note to self: compare to work of Coleman.)

The next point is the following. Suppose one now simply replaces \(X^{\mathrm{ord}}\) by \(X\). Then the cohomology theory \(H^{\bullet}_{c,\partial}\) is probably *literally* computing the Betti cohomology of \(Y\). The Betti cohomology of \(Y\) does indeed see the classes coming from the boundary that we would like to find.

Recall that \(W \setminus Y\) is a normal crossings divisor. Let \(\partial_0\) denote the variety, \(\partial_1\) the (disjoint) union of the irreducible components of the boundary divisor, \(\partial_2\) the union of the intersection of these components, and so on. One now writes down another 1st page spectral sequence as follows:

\(\mathbb{H}^j(\partial_i, \Omega^{\bullet}_{W^{\mathrm{ord}}}(\log \infty)) \Rightarrow H^{i+j}_{c,\partial}(X^{\mathrm{ord}}).\)

This is supposed to be an example of the following: in a nice geometric situation (normal crossings divisor) one may compute cohomology with compact supports in terms of the cohomology of the boundary strata. (I’m still a little confused why \(H^*_{c,\partial}\) is cohomology with compact supports rather than the cohomology of the interior, but anyway…(update: this is explained above: the presence of \(\mathcal{L}\) means it has compact supports in the direction of \(W\setminus Y\), but not \(W \setminus W^{\mathrm{ord}}\))). Moreover, a key point is that the LHS can be interpreted as the rigid cohomology of \(\partial_i\). This allows one to use results of Berthelot and Chiarellotto to deduce that the terms of the LHS are given in terms of the rigid cohomology of (open) varieties. In particular:

1. They admit a theory of weights,
2. \(H^j\) is mixed of weight at least \(j\).
3. They are finite dimensional.

We deduce that RHS is also mixed of weight at least \(i+j\) and finite dimensional.

We want our \(\Pi\) to occur in the RHS, so it certainly suffices to show it actually occurs in \(H^0\). But then by weights it suffices to show that it is coming from the \(H^0\)-terms in the LHS. These are simply given by component groups, and so the computation reduces to a problem concerning the combinatorics of the boundary, on which we shall say more in part III.